TSTP Solution File: SET663+3 by Twee---2.4.2
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%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : SET663+3 : TPTP v8.1.2. Released v2.2.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n012.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 15:32:54 EDT 2023
% Result : Theorem 0.14s 0.45s
% Output : Proof 0.14s
% Verified :
% SZS Type : -
% Comments :
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%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.09 % Problem : SET663+3 : TPTP v8.1.2. Released v2.2.0.
% 0.00/0.10 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.09/0.30 % Computer : n012.cluster.edu
% 0.09/0.30 % Model : x86_64 x86_64
% 0.09/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.30 % Memory : 8042.1875MB
% 0.09/0.30 % OS : Linux 3.10.0-693.el7.x86_64
% 0.09/0.30 % CPULimit : 300
% 0.09/0.30 % WCLimit : 300
% 0.09/0.30 % DateTime : Sat Aug 26 16:22:10 EDT 2023
% 0.09/0.30 % CPUTime :
% 0.14/0.45 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10
% 0.14/0.45
% 0.14/0.45 % SZS status Theorem
% 0.14/0.45
% 0.14/0.46 % SZS output start Proof
% 0.14/0.46 Take the following subset of the input axioms:
% 0.14/0.46 fof(p1, axiom, ![B]: (ilf_type(B, set_type) => (subset(B, empty_set) => B=empty_set))).
% 0.14/0.46 fof(p13, axiom, ![B2]: (ilf_type(B2, set_type) => (ilf_type(B2, binary_relation_type) <=> (relation_like(B2) & ilf_type(B2, set_type))))).
% 0.14/0.46 fof(p2, axiom, ![B2]: (ilf_type(B2, binary_relation_type) => ((domain_of(B2)=empty_set | range_of(B2)=empty_set) => B2=empty_set))).
% 0.14/0.46 fof(p28, axiom, ![B2]: (ilf_type(B2, set_type) => ![C]: (ilf_type(C, set_type) => ![D]: (ilf_type(D, subset_type(cross_product(B2, C))) => relation_like(D))))).
% 0.14/0.46 fof(p3, axiom, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => ![D2]: (ilf_type(D2, relation_type(B2, C2)) => (subset(domain_of(D2), B2) & subset(range_of(D2), C2)))))).
% 0.14/0.46 fof(p33, axiom, ![B2]: ilf_type(B2, set_type)).
% 0.14/0.46 fof(p6, axiom, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => (![D2]: (ilf_type(D2, subset_type(cross_product(B2, C2))) => ilf_type(D2, relation_type(B2, C2))) & ![E]: (ilf_type(E, relation_type(B2, C2)) => ilf_type(E, subset_type(cross_product(B2, C2)))))))).
% 0.14/0.46 fof(prove_relset_1_26, conjecture, ![B2]: (ilf_type(B2, set_type) => ![C2]: (ilf_type(C2, set_type) => ![D2]: (ilf_type(D2, relation_type(B2, C2)) => (ilf_type(D2, relation_type(empty_set, C2)) => D2=empty_set))))).
% 0.14/0.46
% 0.14/0.46 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.14/0.46 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.14/0.46 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.14/0.46 fresh(y, y, x1...xn) = u
% 0.14/0.46 C => fresh(s, t, x1...xn) = v
% 0.14/0.46 where fresh is a fresh function symbol and x1..xn are the free
% 0.14/0.46 variables of u and v.
% 0.14/0.46 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.14/0.46 input problem has no model of domain size 1).
% 0.14/0.46
% 0.14/0.46 The encoding turns the above axioms into the following unit equations and goals:
% 0.14/0.46
% 0.14/0.46 Axiom 1 (p33): ilf_type(X, set_type) = true2.
% 0.14/0.46 Axiom 2 (p2): fresh(X, X, Y) = Y.
% 0.14/0.46 Axiom 3 (p28): fresh46(X, X, Y) = true2.
% 0.14/0.46 Axiom 4 (p1): fresh35(X, X, Y) = empty_set.
% 0.14/0.46 Axiom 5 (p13_1): fresh28(X, X, Y) = ilf_type(Y, binary_relation_type).
% 0.14/0.46 Axiom 6 (p13_1): fresh27(X, X, Y) = true2.
% 0.14/0.46 Axiom 7 (p2): fresh22(X, X, Y) = empty_set.
% 0.14/0.46 Axiom 8 (p1): fresh3(X, X, Y) = Y.
% 0.14/0.46 Axiom 9 (prove_relset_1_26_2): ilf_type(d, relation_type(empty_set, c)) = true2.
% 0.14/0.46 Axiom 10 (p3): fresh90(X, X, Y, Z) = true2.
% 0.14/0.46 Axiom 11 (p28): fresh16(X, X, Y, Z) = relation_like(Z).
% 0.14/0.46 Axiom 12 (p3): fresh13(X, X, Y, Z) = subset(domain_of(Z), Y).
% 0.14/0.46 Axiom 13 (p2): fresh(ilf_type(X, binary_relation_type), true2, X) = fresh22(domain_of(X), empty_set, X).
% 0.14/0.46 Axiom 14 (p6_1): fresh88(X, X, Y, Z, W) = true2.
% 0.14/0.46 Axiom 15 (p28): fresh45(X, X, Y, Z, W) = fresh46(ilf_type(Y, set_type), true2, W).
% 0.14/0.46 Axiom 16 (p13_1): fresh28(relation_like(X), true2, X) = fresh27(ilf_type(X, set_type), true2, X).
% 0.14/0.46 Axiom 17 (p6_1): fresh7(X, X, Y, Z, W) = ilf_type(W, subset_type(cross_product(Y, Z))).
% 0.14/0.46 Axiom 18 (p1): fresh3(subset(X, empty_set), true2, X) = fresh35(ilf_type(X, set_type), true2, X).
% 0.14/0.46 Axiom 19 (p3): fresh89(X, X, Y, Z, W) = fresh90(ilf_type(Y, set_type), true2, Y, W).
% 0.14/0.46 Axiom 20 (p6_1): fresh87(X, X, Y, Z, W) = fresh88(ilf_type(Y, set_type), true2, Y, Z, W).
% 0.14/0.46 Axiom 21 (p3): fresh89(ilf_type(X, relation_type(Y, Z)), true2, Y, Z, X) = fresh13(ilf_type(Z, set_type), true2, Y, X).
% 0.14/0.46 Axiom 22 (p6_1): fresh87(ilf_type(X, relation_type(Y, Z)), true2, Y, Z, X) = fresh7(ilf_type(Z, set_type), true2, Y, Z, X).
% 0.14/0.46 Axiom 23 (p28): fresh45(ilf_type(X, subset_type(cross_product(Y, Z))), true2, Y, Z, X) = fresh16(ilf_type(Z, set_type), true2, Y, X).
% 0.14/0.46
% 0.14/0.46 Goal 1 (prove_relset_1_26_4): d = empty_set.
% 0.14/0.46 Proof:
% 0.14/0.46 d
% 0.14/0.46 = { by axiom 2 (p2) R->L }
% 0.14/0.46 fresh(true2, true2, d)
% 0.14/0.46 = { by axiom 6 (p13_1) R->L }
% 0.14/0.46 fresh(fresh27(true2, true2, d), true2, d)
% 0.14/0.46 = { by axiom 1 (p33) R->L }
% 0.14/0.46 fresh(fresh27(ilf_type(d, set_type), true2, d), true2, d)
% 0.14/0.46 = { by axiom 16 (p13_1) R->L }
% 0.14/0.46 fresh(fresh28(relation_like(d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 11 (p28) R->L }
% 0.14/0.46 fresh(fresh28(fresh16(true2, true2, empty_set, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 1 (p33) R->L }
% 0.14/0.46 fresh(fresh28(fresh16(ilf_type(c, set_type), true2, empty_set, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 23 (p28) R->L }
% 0.14/0.46 fresh(fresh28(fresh45(ilf_type(d, subset_type(cross_product(empty_set, c))), true2, empty_set, c, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 17 (p6_1) R->L }
% 0.14/0.46 fresh(fresh28(fresh45(fresh7(true2, true2, empty_set, c, d), true2, empty_set, c, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 1 (p33) R->L }
% 0.14/0.46 fresh(fresh28(fresh45(fresh7(ilf_type(c, set_type), true2, empty_set, c, d), true2, empty_set, c, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 22 (p6_1) R->L }
% 0.14/0.46 fresh(fresh28(fresh45(fresh87(ilf_type(d, relation_type(empty_set, c)), true2, empty_set, c, d), true2, empty_set, c, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 9 (prove_relset_1_26_2) }
% 0.14/0.46 fresh(fresh28(fresh45(fresh87(true2, true2, empty_set, c, d), true2, empty_set, c, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 20 (p6_1) }
% 0.14/0.46 fresh(fresh28(fresh45(fresh88(ilf_type(empty_set, set_type), true2, empty_set, c, d), true2, empty_set, c, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 1 (p33) }
% 0.14/0.46 fresh(fresh28(fresh45(fresh88(true2, true2, empty_set, c, d), true2, empty_set, c, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 14 (p6_1) }
% 0.14/0.46 fresh(fresh28(fresh45(true2, true2, empty_set, c, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 15 (p28) }
% 0.14/0.46 fresh(fresh28(fresh46(ilf_type(empty_set, set_type), true2, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 1 (p33) }
% 0.14/0.46 fresh(fresh28(fresh46(true2, true2, d), true2, d), true2, d)
% 0.14/0.46 = { by axiom 3 (p28) }
% 0.14/0.46 fresh(fresh28(true2, true2, d), true2, d)
% 0.14/0.46 = { by axiom 5 (p13_1) }
% 0.14/0.46 fresh(ilf_type(d, binary_relation_type), true2, d)
% 0.14/0.46 = { by axiom 13 (p2) }
% 0.14/0.46 fresh22(domain_of(d), empty_set, d)
% 0.14/0.46 = { by axiom 8 (p1) R->L }
% 0.14/0.46 fresh22(fresh3(true2, true2, domain_of(d)), empty_set, d)
% 0.14/0.46 = { by axiom 10 (p3) R->L }
% 0.14/0.46 fresh22(fresh3(fresh90(true2, true2, empty_set, d), true2, domain_of(d)), empty_set, d)
% 0.14/0.46 = { by axiom 1 (p33) R->L }
% 0.14/0.46 fresh22(fresh3(fresh90(ilf_type(empty_set, set_type), true2, empty_set, d), true2, domain_of(d)), empty_set, d)
% 0.14/0.46 = { by axiom 19 (p3) R->L }
% 0.14/0.46 fresh22(fresh3(fresh89(true2, true2, empty_set, c, d), true2, domain_of(d)), empty_set, d)
% 0.14/0.46 = { by axiom 9 (prove_relset_1_26_2) R->L }
% 0.14/0.46 fresh22(fresh3(fresh89(ilf_type(d, relation_type(empty_set, c)), true2, empty_set, c, d), true2, domain_of(d)), empty_set, d)
% 0.14/0.46 = { by axiom 21 (p3) }
% 0.14/0.46 fresh22(fresh3(fresh13(ilf_type(c, set_type), true2, empty_set, d), true2, domain_of(d)), empty_set, d)
% 0.14/0.47 = { by axiom 1 (p33) }
% 0.14/0.47 fresh22(fresh3(fresh13(true2, true2, empty_set, d), true2, domain_of(d)), empty_set, d)
% 0.14/0.47 = { by axiom 12 (p3) }
% 0.14/0.47 fresh22(fresh3(subset(domain_of(d), empty_set), true2, domain_of(d)), empty_set, d)
% 0.14/0.47 = { by axiom 18 (p1) }
% 0.14/0.47 fresh22(fresh35(ilf_type(domain_of(d), set_type), true2, domain_of(d)), empty_set, d)
% 0.14/0.47 = { by axiom 1 (p33) }
% 0.14/0.47 fresh22(fresh35(true2, true2, domain_of(d)), empty_set, d)
% 0.14/0.47 = { by axiom 4 (p1) }
% 0.14/0.47 fresh22(empty_set, empty_set, d)
% 0.14/0.47 = { by axiom 7 (p2) }
% 0.14/0.47 empty_set
% 0.14/0.47 % SZS output end Proof
% 0.14/0.47
% 0.14/0.47 RESULT: Theorem (the conjecture is true).
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